Figure REF  depicts EELS for a \(34\,\) nm Ag film (with no graphene coating) on various substrates: SiO\(_2\) /Si, mono-layer, tri-layer, and 10-layer hBN, and semi-infinite vacuum (as would occur for a suspended sample). Introduction of the high band-gap hBN substrate dramatically enhances surface modes as compared to SiO\(_2\) /Si. The result is striking — the sample with a monolayer hBN substrate has a spectral profile nearly identical to that of a suspended sample. The EELS remain very nearly identical to the suspended case for up to 10 layers. This is in accordance with experimental results that have found the electron mobility of graphene on hBN to be nearly that of suspended graphene. [1]} The similarity of the hBN substrate to the vacuum is expected due to its ultra-wide bandgap, and in the case of our model partly due to the atomic layer thickness. Upon inspection of Fig. REF  (b), it is clear that both real and imaginary parts of hBN's dielectric function are small; therefore at a thickness of \(3.4\,\) Å, the properties of this substrate approach those of the vacuum. The hBN substrate appears to dramatically enhance coupling to the surface plasmon resonance when compared to the SiO\(_2\) /Si substrate, as well as reducing the peak strength of the lossy bulk plasmon.
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